Solve (2x+3)^2-3(x-4)(x+4)-(x-2)^2+1

$$( 2 x + 3 ) ^ { 2 } - 3 ( x - 4 ) ( x + 4 ) - ( x - 2 ) ^ { 2 } + 1$$

Evaluate

$$16x+54$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-\left(x-2\right)^{2}+1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-\left(x^{2}-4x+4\right)+1$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-x^{2}+4x-4+1$$

Add -4 and 1 to get -3.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-x^{2}+4x-3$$

Use the distributive property to multiply -3 by x-4.

$$4x^{2}+12x+9+\left(-3x+12\right)\left(x+4\right)-x^{2}+4x-3$$

Use the distributive property to multiply -3x+12 by x+4 and combine like terms.

$$4x^{2}+12x+9-3x^{2}+48-x^{2}+4x-3$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+12x+9+48-x^{2}+4x-3$$

Add 9 and 48 to get 57.

$$x^{2}+12x+57-x^{2}+4x-3$$

Combine x^{2} and -x^{2} to get 0.

$$12x+57+4x-3$$

Combine 12x and 4x to get 16x.

$$16x+57-3$$

Subtract 3 from 57 to get 54.

$$16x+54$$

Expand

$$16x+54$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-\left(x-2\right)^{2}+1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-\left(x^{2}-4x+4\right)+1$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-x^{2}+4x-4+1$$

Add -4 and 1 to get -3.

$$4x^{2}+12x+9-3\left(x-4\right)\left(x+4\right)-x^{2}+4x-3$$

Use the distributive property to multiply -3 by x-4.

$$4x^{2}+12x+9+\left(-3x+12\right)\left(x+4\right)-x^{2}+4x-3$$

Use the distributive property to multiply -3x+12 by x+4 and combine like terms.

$$4x^{2}+12x+9-3x^{2}+48-x^{2}+4x-3$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+12x+9+48-x^{2}+4x-3$$

Add 9 and 48 to get 57.

$$x^{2}+12x+57-x^{2}+4x-3$$

Combine x^{2} and -x^{2} to get 0.

$$12x+57+4x-3$$

Combine 12x and 4x to get 16x.

$$16x+57-3$$

Subtract 3 from 57 to get 54.

$$16x+54$$